一次称量硬币标签的制作
Confirming the Labels of Coins in One Weighing
摘要：
有装着硬币的$n$包，看起来都一样。每个袋子里有无限多的硬币，同一个袋子里的硬币重量相同。不同袋子里的硬币的重量分别为1、2、3，以此类推，精确到$n$克。每个袋子上都有一个从1到$n$的唯一标签，应该与袋子中硬币的重量相对应。任务是使用天平一次确认所有标签。我们研究称体重的方法，我们称之为下坡法：他们使用袋子里的硬币数量，按递减的顺序。我们展示了这种权衡的重要性。我们在下坡称重中找到最小可能的硬币总重量，以确认袋子上的标签。我们还找到了这样一个加权所需的最小硬币数的界限。
英文摘要：
There are $n$ bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, and so on to $n$ grams exactly. There is a unique label from the set 1 through $n$ attached to each bag that is supposed to correspond to the weight of the coins in that bag. The task is to confirm all the labels by using a balance scale once. We study weighings that we call downhill: they use the numbers of coins from the bags that are in a decreasing order. We show the importance of such weighings. We find the smallest possible total weight of coins in a downhill weighing that confirms the labels on the bags. We also find bounds on the smallest number of coins needed for such a weighing.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
